Kurtosis

Kurtosis is the fourth centralized moment of a probability density function. It is meant to capture the flatness of a distribution. Hence, a small kurtosis implies a distribution that is concentrated around a small range of values of the underlying random variable, while a large kurtosis corresponds to a distribution that is very flat and spread out.

Since the normal distribution has a kurtosis of 3, some analysts prefer to use excess kurtosis, which is defined as kurtosis minus 3 and measures the kurtosis of a particular distribution in excess of that for the normal distribution.

Many option pricing models, such as the Black-Scholes model, assume that asset prices are normally distributed. In practice, this assumption is rarely met and asset prices have distributions with kurtosis higher than the normal distribution, which has important ramifications for option pricing.

Black-Scholes implied volatilities often exhibit a “smile” when plotted against strike price. It has been suggested that one possible explanation for the smile is asset prices that have greater kurtosis than the normal distribution allows.

Kurtosis
Kurtosis
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